Abstract

Suppose that \(\kappa \) is indestructibly supercompact and there is a measurable cardinal \(\lambda > \kappa \) . It then follows that \(A_0 = \{\delta < \kappa \mid \delta \) is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is nonlinear \(\}\) is unbounded in \(\kappa \) . If the Mitchell ordering of normal measures over \(\lambda \) is also linear, then by reflection (and without any use of indestructibility), \(A_1= \{\delta < \kappa \mid \delta \) is a measurable cardinal and the Mitchell ordering of normal measures over \(\delta \) is linear \(\}\) is unbounded in \(\kappa \) as well. The large cardinal hypothesis on \(\lambda \) is necessary. We demonstrate this by constructing via forcing two models in which \(\kappa \) is supercompact and \(\kappa \) exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that \(A_0\) is unbounded in \(\kappa \) if \(\lambda > \kappa \) is measurable. In one of these models, for every measurable cardinal \(\delta \) , the Mitchell ordering of normal measures over \(\delta \) is linear. In the other of these models, for every measurable cardinal \(\delta \) , the Mitchell ordering of normal measures over \(\delta \) is nonlinear.

中文翻译：

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米切尔排序的坚不可摧和线性

摘要

假设\(\kappa \)是不可破坏的超紧，并且存在可测基数\(\lambda > \kappa \)。那么\(A_0 = \{\delta < \kappa \mid \delta \)是一个可测基数，并且\(\delta \)上正常测度的 Mitchell 排序是非线性的\(\}\)是无界的\(\kappa \)。如果\(\lambda \)上正常测度的米切尔排序也是线性的，则通过反射（并且不使用任何不可破坏性），\(A_1= \{\delta < \kappa \mid \delta \)是一个可测的在\(\delta \)上正常测量的基数和米切尔排序是线性的\(\}\)在\(\kappa \)中也是无界的。关于\(\lambda \) 的大基数假设是必要的。我们通过强制构建两个模型来证明这一点，其中\(\kappa \)是超紧凑的，并且\(\kappa \)表现出比完全不可破坏性稍弱的不可破坏性，但足以推断\(A_0\)在\( \kappa \)如果\(\lambda > \kappa \)是可测量的。在其中一个模型中，对于每个可测量的基数\(\delta \) ，正常测量在\(\delta \)上的米切尔排序是线性的。在另一个模型中，对于每个可测量的基数\(\delta \) ，正常测量在\(\delta \)上的米切尔排序是非线性的。